Lesson Content |
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| Similarity and Congruence
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Objective |
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| Students will know how to determine that two figures are similar or congruent by investigating figures that are similar and figures that are congruent. Then they will know how to prove that two figures are similar or congruent by using definitions, postulates, and theorems. |
Other Indicators Addressed |
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2.1.1 The student will analyze the properties of geometric figures.
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Approximate Time |
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| One 30 minute lesson for the discovery lesson and up to 60 minutes for the practice, depending on how well students remember the pre-requisite skills. |
Prerequisite Concepts Needed |
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| Students will need to be able to measure segments using a ruler, measure angles using a protractor, and be familiar with the theorems and postulates used to prove figures similar and congruent. |
Materials Needed |
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Lesson Structure |
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Warm-Up/Opening Activity |
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Use the Prediction Guide to determine what students already know. Feel free to add/delete/change statements as they fit your class.
Accept students' answers and justifications and ask if the class thinks they are reasonable.
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Development of Ideas |
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Worksheet: Comparing Sizes of Figures
Ask students to work in groups of 2 or 3 to measure all of the segments listed using a ruler. It is helpful if all students measure using the same unit (suggest measuring in centimeters). It may be helpful to have the students change their ratios to a decimal (about 1.4) to show that the ratios are equal.
Explorations/Investigations
Present the concept and discover the SSS and SAS postulates of congruency and the SSS, SAS and AA postulates of similarity.
Activity One
Worksheet: Congruent and Similar Triangle Investigation Activity One
Organize students into groups of two. Each student needs to cut out the three strips of paper. Place the strips together corner-to-corner to create a triangle. Compare your triangle to your partner's and to other triangles in the class. Are they congruent?
Encourage the students to prove the triangles congruent using the definition of congruent triangles. Measure each side with a ruler and each angle with a protractor (measuring each side may not be necessary if students realize that everyone started with the same three strips of paper).
What was the minimum amount of information used to create these two congruent triangles? [three congruent sides- this demonstrates SSS theorem of congruency]
Can you replicate this process in the same manner? Explain. [Allow students to create another set of congruent triangles if they are not convinced. You can use the beginning of Activity Two to show this congruence.]
Activity Two
Worksheet: Congruent and Similar Triangle Investigation Activity Two
Only ONE person in each group: Using the three strips of paper from Activity One, fold and cut each strip of paper in half. Create a triangle using the one-half pieces of each of the original strips. Compare this triangle to the first one. Describe any similarities and differences. [the two triangles are not congruent, but are the same shape. These two triangles are similar. You may want to emphasize that all of the NEW triangles are congruent to each other- have students prove this to you using SSS] Help students trace the new triangle and label the sides the same as in Activity One.
Students will justify the similarity of the old and new triangles. Have students measure each side of the triangle and place the measure in the space requested on the worksheet. Note that the ratios of the lengths of each pair of corresponding sides are proportional.
Have students measure each pair of corresponding angles. Note that the measures of each pair of corresponding angles are congruent. The answers in #4 and #5 show that the triangles are similar.
Activity Three
Worksheet: Congruent and Similar Triangle Investigation Activity Three
Each student will copy the two sides and included angle using patty paper. Students should now draw segment AC to create a triangle. Lead a discussion about what parts of the triangle were given and how their measures compare with everyone else's [SAS].
Now, students will compare their triangle to the other triangles in the group. Have students justify that the triangles are congruent – or similar [by definition they are both. Encourage them to show that the corresponding three sides of the triangles are congruent and therefore the triangles are congruent.]
Ask students to locate the midpoint of AB and of BC and label these points D and E, respectively. Draw segment DE. Compare this triangle to triangle ABC. [the two are similar]. Use the definition of similar triangles to justify that the triangles are similar.
Lead a discussion about what parts of the triangle were given and how this is a minimal amount of information needed in order to prove two triangles similar [SAS].
Activity Four
Worksheet: Congruent and Similar Triangle Investigation Activity Four
Students will now trace two angles and use these to create two triangles. These directions may be difficult for students to follow- please try to demonstrate using the overhead and transparencies.
Have the students create two different triangles. Lead a discussion about why the two triangles are not congruent and that AA [or AAA] is not a way to prove two triangles congruent.
The two triangles they create will be similar. Give them time to measure the sides and show that the ratios of the corresponding sides are proportional. This will be a great revelation because the ratios of the corresponding sides will be different than the others they have seen in these activities (all of the former being 2:1). Also have the students show that the three angles are congruent to each other.
Lead a discussion about how the minimum amount of information needed to prove two triangles similar is that two corresponding angles must be congruent. [AA]
By the time they finish these investigations, they will be ready to believe that the ASA theorem of congruence works, or you can show this using a demonstrations. Discuss that ASA works for similarity as well and can really be the same as the AA theorem.
Worksheet: Practice with Congruent and Similar Triangles.
Worksheet: Applications
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Closure |
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| Go back to the Prediction Guide. Allow students to make corrections to their Prediction Guides since they have worked through this lesson. Discuss any corrections and the correct answers. |
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Additional Resources |
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| Please use the HSA public release items to share problems like this with your students. |
/share/clg/xml/lesson_plans/mathematics/SimilarityCongru_221.xml |
